International Journal of Heat and Mass Transfer

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1 nternational Journal of Heat and Mass Transfer 52 (29) Contents lists available at ScienceDirect nternational Journal of Heat and Mass Transfer journal homepage:.elsevier.com/locate/ijhmt Transient radiative heating characteristics of slabs in a alking beam type reheating furnace Sang Heon Han a, *, Seung Wook Baek a, Man Young Kim b a Department of Aerospace, Korea Advanced nstitute of Science and Technology, Guseong-dong, Yuseong-gu Daejeon 35-71, Republic of Korea b School of Mechanical and Aerospace Systems Engineering, Chonbuk National University, Duckjin-dong, Duckjin-gu, Jeonju, Chonbuk , Republic of Korea article info abstract Article history: Received 29 March 27 Received in revised form 14 March 28 Available online 22 September 28 Keyords: Reheating furnace Radiative slab heating Residence time Transient radiative heating characteristics of slabs in a alking beam type reheating furnace is predicted by the finite-volume method (FVM) for radiation. The FVM can calculate the radiative intensity absorbed and emitted by hot gas as ell as emitted by the all ith curvilinear geometry. The non-gray eighted sum of gray gas model (WSGGM) hich is more realistic than the gray gas model is used for better accurate prediction of gas radiation. The block-off procedure is applied to the treatment of the slabs inside hich intensity has no meaning. Entire domain is divided into eight sub-zones to specify temperature distribution, and each sub-zone has different temperatures and the same species composition. Temperature field of a slab is acquired by solving the transient 3D heat conduction equation. ncident radiation flux into a slab is used for the boundary condition of the heat conduction equation governing the slab temperature. The movement of the slabs is taken into account and calculation is performed during the residence time of a slab in the furnace. The slab heating characteristics is also investigated for the various slab residence times. Main interest of this study is the transient variation of the average temperature and temperature non-uniformity of the slabs. Ó 28 Elsevier Ltd. All rights reserved. 1. ntroduction * Corresponding author. Tel.: ; fax: address: freezia@kaist.ac.kr (S.H. Han). One of major concerns regarding a reheating furnace is heating characteristics of slabs. There are to requirements for a slab to satisfy before its exit to the rolling mill. While the temperature of a slab should be elevated up to intended temperature, the temperature gradient inside the slab should remain belo 5 K/m before its exit. The second largest energy is consumed to reheat slabs before rolling process in hot strip mill hile the largest energy is consumed in the process of melting ore. A great deal of attention is thus needed to reduce some parts of unnecessary energy consumption to increase the performance of a furnace. Since a decrease in energy consumption can directly reduce the generation of carbon dioxide, the performance of a reheating furnace is very important. n vie of performance, the shorter residence time of slabs is highly desirable hile maintaining uniform temperature inside slabs. t is ell knon that radiative heat transfer is a major heat transfer mode to heat up slabs inside a reheating furnace. Over 9% heat transfer is via radiation [1,2]. Radiation emitted by hot gas and hot furnace alls reaches to slabs and is then absorbed into slabs. The absorbed radiation is converted into heat and the heat further penetrates into slabs by means of conduction. mportance of radiative heat transfer in heating slabs is ell reflected by the previous study of a reheating furnace. Li et al. [3] calculated radiative heating of a slab in a pusher type furnace chamber using zone method. They used simplified furnace chamber and calculated the slab temperature in to dimension. Anton et al. [4] predicted the influence of the space beteen billets on the productivity of a alking beam type furnace by using Monte Carlo method. The present ork is to simulate the transient radiative heating characteristics of slabs in a OSCO furnace. The finite-volume method (FVM) is used for treating radiation emitted by the alls and gas. The eighted sum of gray gas model (WSGGM) is adopted to take into account the hot gas radiation effect. And the effect of variation of residence time is pursued in this research. The FVM for radiation hich is the same family of the DOM [5 7] as derived by Chui and Chai [8 1] to study thermal radiation in a irregular or curvilinear grid. t has been successfully applied to a problem of to-dimensional curvilinear cavity. Kim and Baek [11] also applied it for analyzing combined conduction, convection and radiation in a gradually expanding channel. They also extended the method to 3D geometry [12]. The FVM can be easily applied to body-fitted coordinates and unstructured mesh [13], since the conservation constraint resulting from integrating the RTE over a control volume and a control angle is satisfied. n this method, the inflo and outflo of radiant energy across control volume faces are balanced ith attenuation and augmentation of /$ - see front matter Ó 28 Elsevier Ltd. All rights reserved. doi:1.116/j.ijheatmasstransfer

2 16 S.H. Han et al. / nternational Journal of Heat and Mass Transfer 52 (29) Nomenclature C specific heat capacity of slab D m ci directional eights f e,k eighting factor used in WSGGM actual radiation intensity (W/(m 2 sr)) b blackbody radiation intensity, =rt 4 /p (W/(m 2 sr)) M total number of radiation direction ~n i unit normal vector at i surface q R all radiation flux (W/m 2 ) ~r position vector (m) ~s unit direction vector T temperature (K) t time (s) x, y, z axes of Cartesian coordinate (m) Greek symbols b extinction coefficient, =j a + r s (m 1 ) DA i, DV surface area and volume of the control volume, respectively D m discrete control angle (sr) e all emissivity q density of slab (kg/m 3 ) h polar angle measured from the z-axis (rad) j a absorption coefficient (m 1 ) k thermal conductivity of a slab (W/(m K)) r Stefan Boltzmann constant, = (W/(m 2 K 4 )) r s scattering coefficient (m 1 ) U phase function of scattering / azimuthal angle measured from the x-axis (rad) W scattering angle beteen ~s and ~s x scattering albedo, =r s /b Superscripts m, n radiation direction o previous time step old previous iteration step R radiation Subscripts E, W, N, S, T, B east, est, north, south, top and bottom neighbors of e,, n, s, t, b east, est, north, south, top and bottom control volume faces k kth gray gas nodal point in hich intensities are located slab slab all radiant energy ithin a control volume and a control angle. Total solid angle, 4p, is discretized into a finite number of discrete solid (control) angle in any convenient manner. 2. Mathematical formulation 2.1. Treatment of slabs Temperature field of a slab is governed by the folloing transient heat conduction equation: qc ot ot ¼ o ox k ot þ o ox oy k ot þ o oy oz k ot ð1þ oz Here q, C, k are mass density, specific heat capacity, and thermal conductivity of a slab, respectively. The incident radiation flux to the slab is used for the boundary condition of Eq. (1): q R slab ¼ ð~r ;~s Þ~s ð ~n Þd ð2þ ¼4p ntegration of Eq. (1) on a cell control volume gives a discretized equation hich relates temperature of the cell ith the temperatures of the neighboring cells: a T ¼ a E T E þ a W T W þ a N T N þ a S T S þ a T T T þ a B T B þ S here ð3aþ a ¼ k ia i ðdxþ i ; ¼ E; W; N; S; T; B; i ¼ e; ; n; s; t; b ð3bþ a ¼ ¼E;W;N;S;T;B S ¼ qcdv T o Dt a þ qcdv Dt ð3cþ ð3dþ n the above equation, subscript denotes a neighboring cell hile i denotes the interfacial surface beteen the current cell and a neighboring cell. (dx) i is the distance beteen the current cell and a neighboring cell. t is assumed that slabs move quick and quiet. On this assumption, it takes no time for slabs to move next positions. So movement of a slab is processed by just transferring temperature data of previous location to those of next location during calculation Finite-volume method for radiation The radiation intensity for gray medium at any position,~r, along a path, ~s through an absorbing, emitting and scattering medium is given by 1 d ð~r;~s Þ b ds ¼ ð~r;~s Þþð1 x Þ b ð~rþþ x 4p ¼4p ð~r;~s ÞU ð~s! ~s Þd ð4þ here b = j a + r s is the extinction coefficient, and x = r s /b is the scattering albedo. U ð~s!~s Þis the scattering phase function for radiation from incoming direction ~s to scattered direction ~s. This equation, if temperature of the medium, b ð~rþ and boundary conditions for intensity are given, provides a distribution of the radiation intensity in medium. The boundary condition for a diffusely emitting and reflecting all can be denoted by ð~r ;~s Þ ¼e b ð~r Þþ 1 e ð~r p ;~s Þ~s j ~n jd ð5þ ~s ~n < here e is the all emissivity and ~n is the unit normal vector at the all. The above equation illustrates that the leaving intensity from the all is a summation of emitted and reflected intensities. The final discretized equation for a general control volume and control angle in three-dimensional geometry can be ritten as mn here ¼ ¼E;W;N;S;T;B ¼ DA i D mn ci;in ¼ i¼e;;n;s;t;b mn þ b mn DA i D mn ci;out þ b o;dvd mn ð6aþ ð6bþ ð6cþ

3 b mn ¼ðb o S mn R Þ DVDmn ð6dþ S mn R D mn ci ¼ð1 x o Þ b þ x o 4p m n U m n!mn d ¼4p / nþ ¼ / n h mþ h m S.H. Han et al. / nternational Journal of Heat and Mass Transfer 52 (29) ð6eþ ð~s ~n i Þd ð6fþ D mn ci;in D ¼ ð~s ~n i Þd ~s ~n i < ð6gþ mn D mn ci;out D ¼ ð~s ~n i Þd ~s ~n i > ð6hþ mn here DA i and ~n i are the surface area and outard unit normal vector at the surface i, respectively. Here is adopted a step scheme in hich a donstream face intensity is set to equal to the upstream nodal value. The discretization procedure and related quantities are easily found in Baek et al. [12]. n this study, the block-off method suggested by Chai et al. [14] is adopted to treat radiatively inactive regions such as slabs and dead zones. Fig. 2 shos a schematic of blocked-off regions. The point A is in a radiatively fully active cell, hereas D and S are in radiatively fully inactive cells. Although calculation is done over the hole domain, only solutions in the active region are meaningful. t must be noted that an additional source term is introduced into Eq. (6a) for block-off treatment as follos: S A ¼ S A C þ SA mn t becomes S A C ¼ and SA ¼ LN (here LN is a large number) in inactive cells, hereas both S A C and SA are equal to zero in active cells. The coefficients of Eqs. (6c) and (6d) are changed such that ¼ DA i D mn b o; S A DVD mn ð8þ b mn i¼e;;n;s;t;b ci;out þ ¼ b o S mn R þ S A C DVD mn ð7þ The folloing conditions are imposed on an active cell, hich is in direct contact ith an inactive cell, as represented by point B in Fig. 2: S A C ¼ DA D mn c DVD mn S A ¼ e b; þ 1 e p m n D m n c D m n ð9þ ð1þ ð11þ here D mn c is the directional eight at the bounding all composed of ~s and ~n. The hole domain is discretized into (N x N y N z ) control volumes. Note that ~n i, DA i and DV have to be calculated carefully according to grid skeness or curvature. Total solid angle, 4p steradians is divided into (N h N / )=M directions, here h is polar angle and / is azimuthal angle, ranging from to p and from to2p as shon in Fig. 1(b), respectively. Boundary condition in Eq. (5) for a diffusely emitting and reflecting all can be discretized as mn ¼ e b; þ 1 e p m n D m n c;in s~n < ~s ~n > ð12þ Besides the all boundary condition, intensity at symmetry planes is also required, for example, at z = z s planes: ðx; y; z s ; h; /Þ ¼ðx; y; z s ; p h; /Þ ð13þ The iterative solution is terminated hen the folloing convergence is attained: h max mn Here mn;old Fig. 1. Schematics of control volume and control angle.. mn;old S Slab mn A Dead one Fig. 2. Schematic of blocked-off regions. i < 1 6 ð14þ is the previous iteration value of mn. Once the intensity field is obtained, the radiative heat flux into all including a slab surface can be estimated as follos: B D

4 18 S.H. Han et al. / nternational Journal of Heat and Mass Transfer 52 (29) q R ¼ ¼4p ð~r ;~s Þ~s ð ~n Þd ¼ mn Dmn c 2.3. Weighted sum of gray gas model (WSGGM) ð15þ Modest [15] has shon that the WSGGM can be used ith any solution method after replacing the non-gray medium by an equivalent small number of gray medium ith constant absorption coefficients: ¼ k ð16þ k 1 d k b ;k ds ¼ k þð1 x ;k Þf e;k b þ x ;k 4p k ð~s ÞU ð~s!~s Þd ¼4p ð17þ f e;k ¼ 1 J j¼1 b e;k;j T j 1 ð18þ here b e,k,j are referred to as the emissivity gas temperature polynomial coefficients as found in Smith et al. [16]. The total intensity can be found by just summing up all the kth gray gas intensities. Eq. (17) is the radiative transfer equation for the kth gray gas ith constant absorption coefficient, ith blackbody intensity b replaced by a eighted intensity f e,k b. This eighted sum of gray gases model takes account of radiative effects by non-gray gases such as CO 2 and H 2 O in gas mixture. More detailed descriptions of WSGGM are easily found in Modest [15]. 3. Results and discussion 3.1. Configuration of the furnace The furnace is very big in its scale and there exist lots of slabs inside the furnace, so it is inevitable to use large size of grid. Moreover, a slab should have appropriate grid spacing, otherise the solution changes ith grid. t is necessary to take relevant grid spacing for a slab. n addition to geometrical grid, solid angle is also discretized, so actual computational array size is increased as much as multiplied by discretization number of solid angle. This is hy the radiation code developed here as parallelized to be run on clustered Cs by using M library. The parallelization makes it possible to overcome deficiency of physical memory as ell as to save computing time. Fig. 3 shos the half section of the reheating furnace. The furnace is symmetric along the y = plane, so calculation is performed on the half section of the furnace. The coordinates system is set to increase the performance of parallelization. The furnace has the dimension of 5.4 m 1.7 m 36. m. A furnace is divided into three zones preheating, heating, and soaking zones. Slabs are mostly heated in the preheating and heating zone. The role of soaking zone is to reduce the slabs temperature distribution in the slabs. Tenty-nine slabs are heating in the furnace and they experience about 2 h heating. The slabs have the dimension of.23 m 4.8 m 1. m and they are located at the elevation of.21 m from the furnace bottom Simulation of the furnace The entire computational domain is divided into eight subzones to specify temperature distribution as shon in Fig. 4. Usually upper region above slabs has higher temperature than loer region. Vertically, the entire domain is divided into to sections ith respect to slabs. Horizontally, it is divided into four sections. Horizontal four sections are formed by spliting preheating zone into to sections, hile the other to zones remain unsplited. Each sub-zone has constant all and gas temperatures as shon in Table 1. The gas temperature is set to the value hich is 2 K higher than the all temperature. The composition of gas is set to be uniform throughout the entire sub-zones H 2 O:.173, CO 2 :.113, O 2 :.15, N 2 :.699 in mass fraction. Fig. 5 shos the computational grid of the furnace. The size of grid is and 1,946,35 cells are formed. Slabs have the grid size of Grid is clustered inside the slabs along the thickness direction. N h and N / solid angle split number have the value of 4 and 12, respectively. Slabs move every 256 s and Y slab insertion preheatingzone (15.m) heatingzone (1.5m) soakingzone (9.3m) Fig. 3. Shape of the furnace Fig. 4. Sub-zones of the entire domain.

5 19 S.H. Han et al. / nternational Journal of Heat and Mass Transfer 52 (29) Table 1 Temperature distribution of eight sub-zones (K) reheating zone Table 2 roperties of slab Heating zone Soaking zone Former Later Upper Wall Gas Loer Wall Gas they reside in the furnace for 7424 s. Total computational time is equal to slab residence time and the physical time step for calculation is 32 s. Computation is performed on a 16 clustered Cs (3 GHz CU). The emissivity of furnace alls is set to be.75. The density of slab is 7854 kg/m3 hile the other properties of slab are listed in Table 2. Slabs are introduced into the furnace at 293 K, i.e., clod charging of slabs. Fig. 6 depicts radiation flux vector on J = 4 slice. Radiation emitted by hot gas impinges on the furnace all and the slabs. Temperature difference is a potential for driving radiation flux. So large radiation flux is produced beteen hot gas and slabs and large heat flux vector is formed around the slabs. Because the temperature difference beteen slabs and hot gas is bigger than that of beteen the furnace alls and hot gas, most of the radiation flux from hot gas flos into the slabs. Fig. 7 shos three integrated incident radiation fluxes into the slabs on all the surfaces, on the top surface, and on the bottom surface of each slab. The slabs in the preheating and heating zones receive large portion of radiation flux. Among the slabs, eighth slab receives largest radiation flux. Radiation flux becomes eak in soaking zone. t is because slabs already have experienced enough heating in previous zones and the temperature difference becomes very small. Moreover, radiation flux is coming out from bottom of the slabs in the soaking zone. When a slab is heated up, its corners are heated faster than any other region. Three surfaces adjoin on each corner of the slab, here heat penetrates into the slab in three directions. This is compared to the only one direction into the plain surface region. Fig. 8 confirms this phenomenon. Highest temperature spots are located on the corners of the slab and loer temperature zones are formed in the inner area of all the surfaces. Large temperature difference inside a slab is mainly caused by this fast heating of corners. Another main cause of temperature distribution is a skid system but it is not considered in this study. Skid system keeps temperature lo in the contact area beteen a slab and a skid system by shielding radiation flux. Fig. 9 shos the volume averaged temperature, the area averaged temperature on the bottom and top surfaces, and maximum temperature difference inside a slab. The slabs are emitted to rolling mill at about 1273 K. n the soaking zone, slabs stay at almost same temperature but the temperature difference continuously decreases until a slab reaches 26th position. From the 26th Temperature (K) Conductivity (W/m K) Specific heat (J/kg K) Emissivity T < < T < < T < < T < < T < < T position, the decreasing rate of temperature difference reduces to small quantity. Top surface temperature alays stays higher than the bottom surface temperature. But a reversion occurs beteen bottom surface and volume mean temperatures at 22nd slab. Since then, loest temperature is located on the bottom surface and conductive heat flux directs donard, hile conductive heat flux directs inard before then. Mean temperature of a slab should stay at target temperature and temperature difference should stay belo 5 K/m. This requirement is satisfied from 26th slab Variation of residence time Calculation is performed to investigate the appropriateness of the heating time. Three more cases of residence time 567, 6496, and 696 s are tried for the inspection. n previous section, a slab satisfies the requirement from 26th position, so there is a possibility to increase the performance. Target temperature is to be 1273 K because slab temperature converges to it under the given furnace all and gas temperature. t is turned out that the residence times of 5568 and 6496 s are not appropriate. Fig. 1 shos that temperature difference remains over 5 K all the heating time for the cases of 5568 and 6496 s. The case of 696 s satisfies the temperature difference requirement from the 28th slab and reaches close to the target temperature. Further a slab moves into the hot gas field, the higher the radiation flux to the slab gros. Fig. 11 shos that the radiation flux increases after insertion. But radiation flux does not continue to increase because the temperature rise of a slab derives the radiation flux to decrease. So there exists a location at hich maximum radiation flux occurs. t occurs at the ninth slab for the residence times of 5568 and 6496 s. The cases of 696 and 7424 s have the maximum at the eighth slab. The radiation flux turns to decreases right after reaching maximum due to rapid temperature rise of slabs in case of 7424 s. But relatively ide peak is formed in case of 5568 s. The result shos that the increased potential by slab s further entrance into hot gas field enough compensates for the temperature rise of a slab for the case of 5568 s. The radiation fluxes of all the cases are almost equal to each other before reaching their maximum value. This means that the temperature of a slab is not a deterministic factor of the radiation flux in the initial stage of heating for cold charging of slabs. Y Fig. 5. Computational grid.

6 11 S.H. Han et al. / nternational Journal of Heat and Mass Transfer 52 (29) : 1 kw/m 2 Y Fig. 6. Radiation flux vector on J = 4 slice. Radiation Flux(kW/m 2 ) Total Top Surface Bottom Surface Temperature(K) Mean Top Surface Bottom Surface Maximum Difference Heating Soaking [ reheating one ] [ one ] [ one ] Fig. 7. Radiation fluxes into the slabs Fig. 9. Various temperature profiles (K) (b) bottom surface (c) upper surface (d) left side (a) full (e) right side Fig. 8. Temperature contour of the 1th slab.

7 S.H. Han et al. / nternational Journal of Heat and Mass Transfer 52 (29) Temperature(K) Radiation Flux(kW/m 2 ) Conclusion 5568 s 6496 s 696 s 7424 s Mean Temperature Maximum Difference Fig. 1. Temperature profiles for the various residence times Fig. 11. Radiation fluxes for the various residence times s 6496 s 696 s 7424 s A numerical simulation as performed to investigate the radiative heating characteristics of the slabs in a reheating furnace of OSCO incorporation. The radiation flux to each slab inside the furnace is calculated by using parallelized FVM radiation code and it is used for the boundary condition of the heat conduction equation governing slab temperature. recise observation of heating characteristics as carried for the given slab residence time and the effects of slab residence time on heating slabs ere investigated. t is observed that most radiation flux occurs at preheating zone and heating zone. n soaking zone, slabs receive little radiation flux from the environment, so slab temperature is hardly raised but degree of temperature uniformity is further developed. The eighth slab receives largest radiation flux for the given residence time (7424 s). Of the four cases of residence time, the residence times of 696 and 7424 s satisfy the temperature uniformity requirement. The temperature uniformity requirement is satisfied from the 26th slab in case of 7424 s and from the 28th slab in case of 696 s. Their emission temperatures reach close to the target temperature of 1273 K. Acknoledgements This ork as supported by the Combustion Engineering Research Center at the Department of Mechanical Engineering, Korea Advanced nstitute of Science and Technology, hich is founded by the Korea Science and Engineering Foundation. References [1] R. Ford, N.V. Suryanarayana, J.H. Johnson, Heat transfer model for solid-slab/ ater-cooled skid pipe in reheat furnace, ronmak. Steelmak. 7 (198) [2] S.H. Han, S.W. Baek, S.H. Kang, C.Y. Kim, Numerical analysis of heating characteristics of a slab in a bench scale reheating furnace, nt. J. Heat Mass Transfer 5 (27) [3] ongyu Li,.V. Barr, J.K. Brimacombe, Computer simulation of the slab reheating furnace, Can. Metall. Quart. 27 (1988) [4] A. Jaklic, T. Kolenko, B. upancic, The influence of the space beteen the billets on the productivity of a continuous alking-beam furnace, Appl. Therm. Eng. 25 (25) [5] M.F. Modest, Radiative Heat Transfer, McGra-Hill, Ne York, [6] B.G. Carlson, K.D. Lathrop, Transport Theory The Method of Discrete Ordinated in Computing Methods in Reactor hysics, Gordon & Breach Science ublishers, Ne York, [7] W.A. Fiveland, Discrete ordinates solutions of transport equation for rectangular enclosure, ASME J. Heat Transfer 16 (1984) [8] E.H. Chui, G.D. Raithby, Computation of radiant heat transfer on a nonorthogonal mesh using the finite-volume method, Numer. Heat Transfer B 23 (1993) [9] J.C. Chai, H.S. Lee, S.V. atankar, Finite volume method for radiation heat transfer, J. Thermophys. Heat Transfer 8 (1994) [1] J.C. Chai, A Finite-Volume Method for Radiation Heat Transfer, h.d. Thesis, University of Minnesota, Minneapolis, MN, [11] M.Y. Kim, S.W. Baek, Numerical analysis of conduction, convection, and radiation in a gradually expanding channel, Numer. Heat Transfer A 29 (1996) [12] S.W. Baek, M.Y. Kim, J.S. Kim, Nonorthogonal finite-volume solutions of radiative heat transfer in a three-dimensional enclosure, Numer. Heat Transfer B 34 (1998) [13] J.Y. Murthy, S.R. Mathur, Finite volume method for radiative heat transfer using unstructured meshes, J. Thermophys. Heat Transfer 12 (1998) [14] J.C. Chai, H.S. Lee, S.V. atankar, Treatment of irregular geometries using a Cartesian coordinates finite-volume radiation heat transfer procedure, Numer. Heat Transfer B 26 (1994) [15] M.F. Modest, The eighted-sum-of-gray-gases model for arbitrary solution methods in radiative transfer, ASME J. Heat Transfer 113 (1991) [16] T.F. Smith,.F. Shen, J.N. Friedman, Evaluation of coefficients for the eighted sum of gray gases model, ASME J. Heat Transfer 14 (1982)